I've seen this issue only once before and it turned out to be a half-clogged radiator that was the culprit. It did the same thing yours is doing except that this car would also overheat if stuck in traffic or if it sat too long after driving.
When you flushed your system was the coolant clear and looked to be in good condition or was it cloudy or dark? What I believe happened with the car that I had was that someone tried to use some kind of stop leak but didn't do it properly and so it basically sealed a bunch of the slats in the radiator instead.
I found this out by letting the car warm-up and then gently touching the radiator slats. Some were much, much cooler than the others (the ones that were clogged).
As for the engine not starting, is that where it's at now? It will not start at all? Does the starter click but do nothing else?
The reason I ask is because if the engine has been getting hot it is possible that you've blown a head gasket and filled a cylinder with coolant (which, since you can't compress water, will cause the engine to seize-up). An easy way to check for this is to pull the spark plugs, examine each (looking for water or coolant), and try spinning the engine over without the plugs.
Hope that's not the case!
Good luck!